Characterisation of the distribution of random variables in linear structural relations

Abstract

The paper examines the nature of the latent random variables which occur in linear structural models, under the assumption that they are independent. It is shown that alternative representations with different sets of coefficients for the latent variables or different numbers of latent variables are possible when and only when some of the latent variables or some of their linear combinations have a univariate normal distribution. The investigation has applications in the problem of identifiability of parameters in linear structural relations. An important characterization of the multivariate normal distribution is deduced.